ON THE TRIGONOMETRY OF THE PARABOLA. 



Fig. 7. 

 H 



91 



XXXI. There are some curious analogies between the parabola and the 

 circle, considered under this point of view. 



In the parabola, the points T, T,, T,„ which divide the bnes 

 m(secd+t2Lud), m[sec (0-^0) + tan (0^0)] 

 into their component parts, are upon tangents to the parabola. The corre- 

 sponding points B, B„ B, in the circle are on the circumference of the circle. 

 In the parabola, the extremities of the lines m(sec d + tan 0) are on a right 

 line VT • in the circle, the extremities of the bent lines are all in the pomt A. 

 The analogy between the expressions for parabolic and circular arcs will 

 be seen by pStting the expressions under the following forms :— 

 Parabolic arc — log (sec 0+ tan 0) - subtangent =0, 

 Circular arc + log (cos 0+ ^/~l sin 0)^~i- subtangent =0. . (37) 

 The locus of the point T, the intersections of the tangents to the parabola 

 with the perpendiculars from the focus, is a right line ; or in other words, 

 while one end of a subtangent rests on the parabola, the other end rests on a 

 right line. So in the circle ; while one end of the subtangent rests on the 

 circle the other end rests on a cardioide, whose diameter is equal to that ot 

 the circle, and whose cusp is at S. SPA is the cardioide. ^ . ^ ^ „ 

 The length of the tangent VN to any point N is »w(sec + tan 0) = 2wi tan«, 

 when is very large. The length of the cardioide is 2D sin 3-. 



XXXII The radius vector of a circle whose radius is r, drawn from any pomt 

 on the circumference, and making the angle with a diameter drawn through 

 this point, is given by the equation p=2r cos 0, and since the coinciding per- 

 pendicular from this point as focus on a tangent to a parabola ^sp=m sec 0, 

 it follows that pp=2mr, a constant quantity. Hence tlie curves are polar 

 reciprocals one of the other. The circumference of the circle passes through 

 the focus of the parabola. . ,. , , i 



The centre of the circle is the pole of the directrix of the parabola. 

 As the extremities N of all the numbers measured along the sccdar are on 

 a right line VN, the reciprocals of these points will all pass through the 

 point A, the pole of the scalar VN. 



